Which expression is equivalent to [tex]\(\sqrt{\frac{225}{625} m^4 n^6}\)[/tex]?

A. [tex]\(\frac{1}{20} m^2 \left|n^3\right|\)[/tex]
B. [tex]\(\frac{1}{20} m^2 n^4\)[/tex]
C. [tex]\(\frac{3}{5} m^2 \left|n^3\right|\)[/tex]
D. [tex]\(\frac{3}{5} m^2 n^4\)[/tex]



Answer :

Sure! Let's solve the expression step by step.

We need to find an equivalent expression for [tex]\(\sqrt{\frac{225}{625} m^4 n^6}\)[/tex].

1. Separate the constant and variable parts:

The given expression can be written as:

[tex]\[\sqrt{\frac{225}{625} \cdot m^4 \cdot n^6}\][/tex]

2. Simplify the constant part:

The fraction [tex]\(\frac{225}{625}\)[/tex] can be simplified:

[tex]\[\frac{225}{625} = \frac{15^2}{25^2} = \left(\frac{15}{25}\right)^2\][/tex]
[tex]\[\frac{15}{25} = \frac{3}{5}\][/tex]

Thus, we have:

[tex]\[\sqrt{\frac{225}{625}} = \sqrt{\left(\frac{3}{5}\right)^2} = \frac{3}{5}\][/tex]

3. Simplify the variable part:

- For [tex]\(m^4\)[/tex]:

[tex]\[\sqrt{m^4} = m^{4/2} = m^2\][/tex]

- For [tex]\(n^6\)[/tex]:

[tex]\[\sqrt{n^6} = n^{6/2} = n^3\][/tex]

4. Combine the simplified parts:

Now that we have simplified both the constant and variable parts, we combine them:

[tex]\[ \sqrt{\frac{225}{625} m^4 n^6} = \frac{3}{5} \cdot m^2 \cdot |n^3| \][/tex]

5. Comparison with provided options:

- [tex]\(\frac{1}{20} m^2 \left|n^3\right|\)[/tex]
- [tex]\(\frac{1}{20} m^2 n^4\)[/tex]
- [tex]\(\frac{3}{5} m^2 \left|n^3\right|\)[/tex]
- [tex]\(\frac{3}{5} m^2 n^4\)[/tex]

The correct expression that matches our simplified result is:

[tex]\[ \frac{3}{5} m^2 |n^3| \][/tex]

Thus, the equivalent expression is:

[tex]\(\boxed{\frac{3}{5} m^2\left|n^3\right|}\)[/tex]